Evaluate the following iterated integrals. int_1^2int_0^1(3x^2+4y^3)dydx

Evaluate the following iterated integrals. int_1^2int_0^1(3x^2+4y^3)dydx

Question
Integrals
asked 2021-01-10
Evaluate the following iterated integrals.
\(\int_1^2\int_0^1(3x^2+4y^3)dydx\)

Answers (1)

2021-01-11
To evaluate the integral: \(\int_1^2\int_0^1(3x^2+4y^3)dydx\)
Solution:
When we integrate with respect to one variable then other will be kept as constant.
Evaluating the integral.
\(\int_1^2\int_0^1(3x^2+4y^3)dy=\int_1^2[\int_0^1(3x^2+4y^3)dy]dx\)
\(=\int_1^2 [(3x^2y+4\cdot\frac{y^4}{4})_0^1]dx\)
\(=\int_1^2 [3x^2\cdot 1+1^4-0]dx\)
\(=\int_1^2 (3x^2+1)dx\)
\(=[3\cdot\frac{x^3}{3}+x]_1^2\)
\(=[x^3+x]_1^2\)
\(=[(2^3+2)-(1^3+1)]\)
\(=[12-2]\)
\(=10\)
\(\text{Hence, required answer is 10}\)
0

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